Parallel lines are important concepts in axiomatic geometry. The parallel axiom of Euclidean geometry can be expressed as "only one straight line is parallel to the known straight line at a point outside the straight line". However, its negative form "there is no straight line parallel to the known straight line at a point outside the straight line" or "there are at least two straight lines parallel to the known straight line at a point outside the straight line" can be used as a substitute for the parallel axiom of Euclidean geometry, and non-Euclidean geometry independent of Euclidean geometry can be deduced.
If both lines are parallel to the third line, then the two lines are also parallel to each other. If a∨b, b∨c, then a ∨ c.
Parallel lines have the following properties
1. After passing a point outside the straight line, one and only one straight line is parallel to the known straight line.
2. Two parallel lines are cut by a third straight line, with equal congruent angles and equal internal angles, which complement each other.
3. When two lines are parallel to the third line, the two lines are parallel.
4. Parallel lines are proportional to the corresponding sides of the triangle.
These propositions depend on the fifth postulate (parallel axiom) of Euclidean geometry, so they are not valid in non-Euclidean geometry.